Whisk flour and salt in a large bowl. Cut in cold butter and shortening with a pastry cutter until the mixture looks like coarse crumbs with pea sized pieces of fat remaining.
Add ice water one tablespoon at a time, stirring gently after each addition until the dough just comes together when pressed. Shape into a disc, wrap in plastic, and refrigerate for at least 30 minutes.
While the dough chills, combine diced apples, water, granulated sugar, brown sugar, cinnamon, and nutmeg in a medium saucepan. Cook over medium heat until apples soften, about 8-10 minutes.
Stir cornstarch into lemon juice to form a slurry, then stir into the apple mixture and cook 2 more minutes until thickened. Remove from heat and cool completely to room temperature.
Divide the chilled dough in half. On a lightly floured surface, roll one half out to about 1/8 inch thick. Cut into 5-inch circles, rerolling scraps as needed.
Spoon about 2 tablespoons of cooled filling onto one half of each circle, leaving a half inch border. Moisten edges with water, fold over to form a half moon, and press to seal. Crimp with a fork.
Pour 2 inches of vegetable oil into a cast iron skillet or Dutch oven. Heat to 350°F using a deep fry thermometer. Maintain temperature throughout frying.
Using a spider skimmer, gently lower 2-3 pies into the hot oil. Fry for 2-3 minutes per side until deep golden brown and crisp. Drain on a wire rack lined with paper towels.
Let the oil return to 350°F between batches. While still warm, dust the fried pies generously with powdered sugar or cinnamon sugar. Let cool 5 minutes before serving.